Looking for Bird Nests: Identifying Stay Points with Bounded Gaps

A stay point of a moving entity is a region in which it spends a significant amount of time. In this paper, we identify all stay points of an entity in a certain time interval, where the entity is allowed to leave the region but it should return within a given time limit. This definition of stay points seems more natural in many applications of trajectory analysis than those that do not limit the time of entity's absence from the region. We present an $O(n \log n)$ algorithm for trajectories in $R^1$ with $n$ vertices and a $(1 + \epsilon)$-approximation algorithm for trajectories in $R^2$ to identify all such stay points. Our algorithm runs in $O(kn^2)$, where $k$ depends on $\epsilon$ and the ratio of the duration of the trajectory to the allowed gap time.